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Chapter 7: Problem 29

Factor by grouping. $$9 x^{2}-13 x y+4 y^{2}$$

Short Answer

Expert verified
The factored expression by grouping is \((x-y)(9x-4y)\).

Step by step solution

01

Split the middle term

First, we need to split the middle term, -13xy, into two terms that when added will give us -13xy and when multiplied will give us the product of the first and third terms (9x^2 and 4y^2). \(9x^{2}-13xy+4y^{2}\Rightarrow 9x^{2}-9xy-4xy+4y^{2}\) Notice that we have split -13xy into -9xy and -4xy.
02

Group terms

Now, we will group the expression into pairs that have the GCF: \((9x^{2}-9xy)+(-4xy+4y^{2})\)
03

Factor out the GCF from each group

Factor out the GCF from the first group and from the second group separately: 9x is the GCF of the first group, and 4y is the GCF of the second group: \(9x(x-y)+4y(-x+y)\) Notice that we have (x-y) and (-x+y) as factors in both groups. Since (-x+y) is the same as (x-y) multiplied by -1, we can rewrite our expression to have the same binomial in both terms: \(9x(x-y)-4y(x-y)\)
04

Factor out the common binomial

Now, we can factor out the common binomial (x-y) from both terms. This leaves us with: \((x-y)(9x-4y)\) So, the factored form of the given expression is: \((x-y)(9x-4y)\)

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Most popular questions from this chapter

Write an equation and solve. A 13 -ft ladder is leaning against a wall. The distance from the top of the ladder to the bottom of the wall is \(7 \mathrm{ft}\) more than the distance from the bottom of the ladder to the wall. Find the distance from the bottom of the ladder to the wall.

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